D solve partial differential equations second order pdes

LaplaceEquation = D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] == 0;

DSolve[LaplaceEquation, u[x, y], {x, y}]

{{u[x, y] -> C[1][I*x + y] + C[2][-I*x + y]}}

a = 3;b = 1;c = 5;b ^ 2 - 4a * c

-59

eqn = a * D[u[x, y], {x, 2}] + b * D[u[x, y], x, y] + c * D[u[x, y], {y, 2}] == 0;

sol = DSolve[eqn, u, {x, y}]

{{u -> Function[{x, y}, C[1][((-1 + I*Sqrt[59])*x)/6 + y] + C[2][((-1 - I*Sqrt[59])*x)/6 + y]]}}

eqn /. sol//Simplify

{5*Derivative[0, 2][Function[{x, y}, C[1][((-1 + I*Sqrt[59])*x)/6 + y] + C[2][((-1 - I*Sqrt[59])*x)/6 + y]]][x, y] + Derivative[1, 1][Function[{x, y}, C[1][((-1 + I*Sqrt[59])*x)/6 + y] + C[2][((-1 - I*Sqrt[59])*x)/6 + y]]][x, y] + 3*Derivative[2, 0][Function[{x, y}, C[1][((-1 + I*Sqrt[59])*x)/6 + y] + C[2][((-1 - I*Sqrt[59])*x)/6 + y]]][x, y] == 0}

WaveEquation = D[u[x, t], {x, 2}] - D[u[x, t], {t, 2}] == 0;

DSolve[WaveEquation, u[x, t], {t, x}]

DSolve[-Derivative[0, 2][u][x, t] + Derivative[2, 0][u][x, t] == 0, u[x, t], {t, x}]

a = 2;b = 7;c = -1;b ^ 2 - 4a * c

57

eqn = a * D[u[x, y], {x, 2}] + b * D[u[x, y], x, y] + c * D[u[x, y], {y, 2}] == 0;

sol = DSolve[eqn, u, {x, y}]

{{u -> Function[{x, y}, C[1][((-7 - Sqrt[57])*x)/4 + y] + C[2][((-7 + Sqrt[57])*x)/4 + y]]}}

eqn /. sol//Simplify

{-Derivative[0, 2][Function[{x, y}, C[1][((-7 - Sqrt[57])*x)/4 + y] + C[2][((-7 + Sqrt[57])*x)/4 + y]]][x, y] + 7*Derivative[1, 1][Function[{x, y}, C[1][((-7 - Sqrt[57])*x)/4 + y] + C[2][((-7 + Sqrt[57])*x)/4 + y]]][x, y] + 2*Derivative[2, 0][Function[{x, y}, C[1][((-7 - Sqrt[57])*x)/4 + y] + C[2][((-7 + Sqrt[57])*x)/4 + y]]][x, y] == 0}

a = 3;b = 30;c = 75;b ^ 2 - 4a * c

0

eqn = a * D[u[x, y], {x, 2}] + b * D[u[x, y], x, y] + c * D[u[x, y], {y, 2}] == 0;

sol = DSolve[eqn, u, {x, y}]

{{u -> Function[{x, y}, C[1][-5*x + y] + x*C[2][-5*x + y]]}}

eqn /. sol//Simplify

{75*Derivative[0, 2][Function[{x, y}, C[1][-5*x + y] + x*C[2][-5*x + y]]][x, y] + 30*Derivative[1, 1][Function[{x, y}, C[1][-5*x + y] + x*C[2][-5*x + y]]][x, y] + 3*Derivative[2, 0][Function[{x, y}, C[1][-5*x + y] + x*C[2][-5*x + y]]][x, y] == 0}